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Unformatted text preview: ME375 Handouts Thermal Systems
• Basic Modeling Elements
– Resistance
• Conduction
• Convection
• Radiation – Capacitance • Interconnection Relationships
– Energy Balance  1st Law of Thermodynamics • Derive Input/Output Models
School of Mechanical Engineering
Purdue University ME375 Thermal Systems  1 Key Concepts
• q : heat flow rate [J/sec = W]
• T : temperature [oK] or [oC] (
( )
) Temperature in a body usually depends on spatial as well as temporal coordinates. As a
temporal
result, the dynamics of a thermal system has to be described by partial differential equations.
Moreover, nonlinearities are often essential in describing the heat transfer by radiation and
heat
convection. However, very few nonlinear PDEs have analytical (closed form) solutions.
Usually, finite element methods (FEM) are used to numerically solve nonlinear PDE
solve
problems. Our purpose is to try to use lumped model approximations of thermal systems to
approximations
obtain linear ODEs that are capable of describing the dynamic response of thermal systems
to a good first approximation.
For many thermal system, an equilibrium condition exists that defines the nominal operating
defines
condition. In these cases, the deviation of the heat flow rate and temperature from their
nominal values, q and T , are of interest. Thus, we can define the incremental heat flow
define
rate ( q ( t ) ) and the incremental temperature ( T ( t ) ) to be:
q (t ) = q (t ) − q
and
T (t ) = T (t ) − T
School of Mechanical Engineering
Purdue University ME375 Thermal Systems  2 1 ME375 Handouts Basic Modeling Elements
• Thermal Resistance
Ex:
Ex:
Describes the heat transfer process
through an element with the
characteristic that the heat flow rate
across the element is proportional to
the temperature difference across the
element, i.e.
T1 q T2 + ΔT − T1 T2
q R Two bodies at temperatures T1 and T2 are
separated by two elements with different
thermal resistance R1 and R2. Heat flows
through the two elements at a rate of q. Find
the equivalent thermal resistance Req and
solve for the interface temperature between
the two elements. T1 R1 R2 T2 T1 R T2 Req q ΔT = T12 = T1 − T2 = R ⋅ q
or
1
1
q = ( T1 − T2 ) = ΔT
R
R q [oK/W] R= School of Mechanical Engineering
Purdue University ME375 Thermal Systems  3 Three Types of Heat Transfer
• Conduction
Ex: Calculate the equivalent thermal
Ex:
resistance of a wall with a window.
Heat transfer through solid or continuous
Wall
Window
media via random molecular motion
AG
Area
AW
(diffusion).
Thickness
T1 q T2 d T α dW αW dG αG Cross sectional area A T1
T2 q= αA αA x (T1 − T2 ) =
T12 ⇒ R =
d
d
– α : thermal conductivity [W/moK]
School of Mechanical Engineering
Purdue University ME375 Thermal Systems  4 2 ME375 Handouts Three Types of Heat Transfer
• Convection
Heat transfer between the interface of a
solid material and a fluid material via
bulk motion of the fluid.
T
TF q = hA ⋅ (TS − TF ) = hA ⋅ ΔT
x q – A : surface area [m2]
– h : convective heat transfer
coefficient [W/m2oK]
– TS : surface temperature [oK]
– TF : fluid temperature [oK] T
TF
TF x 1
hA
– h depends on surface geometry,
fluid flow rate, temperature, flow
direction, ...
⇒ R= School of Mechanical Engineering
Purdue University ME375 Thermal Systems  5 Three Types of Heat Transfer
• Radiation
Except for radiation, both conductive
Heat transfer via electromagnetic waves. and convective heat transfer processes
T2
can be modeled as thermal
T1
q
resistances.
In the previous discussions, the
Surface Area A
assumption is that the materials do not
4
4
store thermal energy.
In reality,
q = σ FE FV A ⋅ ( T1 − T2 )
materials do store a certain amount of
– A : surface area [m2]
thermal energy.
– σ : StefanBoltzmann constant
Stefan[W/m2oK4]
Q: How would we model the process of
– FE : effective emissivity
storing thermal energy ?
– FV : view factor
Nonlinear! Will not be considered in this
course
School of Mechanical Engineering
Purdue University ME375 Thermal Systems  6 3 ME375 Handouts Basic Modeling Elements
• Thermal Capacitance
The ability of a substance to hold or
store heat is the heat capacity of the
material and it behaves like a thermal
capacitance. Since the specific heat cP
can be interpreted as the heat storage
capacity of the material per unit mass,
the total heat storage capability of a
material is: cP ⋅ M If there is net heat flow into the
material, the temperature of the
material will change and the rate of
temperature change is proportional to
the net heat flow rate qSTORE :
d
cP M TC = qSTORE = qIN − qOUT
dt
We can define the thermal capacitance C = cP M = cP ρ V + TC −
TC qIN C qOUT qIN − qOUT C Mass, M
Volume, V
Density, ρ Note: The above relationship holds only if
Note:
we assume that the temperature is uniform
across the entire material. School of Mechanical Engineering
Purdue University ME375 Thermal Systems  7 Interconnection Laws
• Energy Balance  1st Law of Thermodynamics
– Energy stored in the system is the sum of the net energy inflow, the energy
generated within the system and the work done on the system: q STORE = ∑q IN − ∑q OUT + q GENERATED ' +
W ITHIN d
WW ORK =
dt DONE Ex: A material with a thermal capacitance C is surrounded by an insulation material with
Ex:
thermal resistance R. Heat is added to the inner material at a rate of qi(t). Find the
system model, if the inner material temperature TC is to be the output.
Ta
TC , C
R qi(t)
School of Mechanical Engineering
Purdue University ME375 Thermal Systems  8 4 ME375 Handouts In Class Example
Ex: The Pentium II processor under normal operation will generate heat at a rate of qi(t). The processor
heat
itself has a specific heat of cP. The cross sectional area of the chip is AP with a thickness of dP. The
average density of the processor is ρP . To help dissipate the heat and reduce the processor
temperature TP, a heat sink with the same cross sectional area and an average thickness of dS is added
on top of the processor. The heat sink has a thermal conductivity of αS . To further improve heat
conductivity
dissipation, a fan is used to generate air flow on top of the heat sink, the effective convection
heat
coefficient is hA and the effective contact area between the heat sink and the air flow is AS . The
air
temperature inside the computer is maintained at TA. Find the relationship between qi(t) and the
temperature of the processor TP.
TA
hA
AP Heat Sink
cP , ρP , TP School of Mechanical Engineering
Purdue University dS
dP ME375 Thermal Systems  9 5 ...
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This note was uploaded on 08/28/2010 for the course ME 375 taught by Professor Meckle during the Spring '10 term at Purdue.
 Spring '10
 Meckle
 Mechanical Engineering

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